# Automated Theorem Proving

In Lean 4 theorem proving that depends on project context, failed proofs used to be discarded as a final filter result, and independent sampling restarted later attempts nearly from scratch. This search framework draws one candidate each from a general-purpose reasoning model and a Lean-specialized prover, uses compiler errors in a pairwise comparison to pick the current-best proof, accepts a revision only when the comparison judges it better, and resamples when refinement stagnates. On seven real Lean 4 projects from miniCTX-v2, within the pass@32 budget it improves average pass rate by 12.8 percentage points over the two base models' average pass@32 while cutting LLM calls by 21.9%.

In Lean 4 theorem proving that depends on project context, failed proofs used to be discarded as a final filter result, and independent sampling restarted later attempts nearly from scratch. This search framework draws one candidate each from a general-purpose reasoning model and a Lean-specialized prover, uses compiler errors in a pairwise comparison to pick the current-best proof, accepts a revision only when the comparison judges it better, and resamples when refinement stagnates. On seven real Lean 4 projects from miniCTX-v2, within the pass@32 budget it improves average pass rate by 12.8 percentage points over the two base models' average pass@32 while cutting LLM calls by 21.9%.

When a general-purpose model writes a geometry proof, it first retrieves structurally similar solved problems as examples, and a symbolic verifier then points out errors round by round. On 50 SAT-level Euclidean geometry problems from FormalGeo-7k, the full pipeline reaches 68%–96% proof accuracy, against 10%–44% for baselines that use neither analogical retrieval nor verifier feedback. Under the same verifier feedback and inference budget, analogical guidance alone lifts GPT-5 from 88% to 96% and o1 from 52% to 80%.

When a general-purpose model writes a geometry proof, it first retrieves structurally similar solved problems as examples, and a symbolic verifier then points out errors round by round. On 50 SAT-level Euclidean geometry problems from FormalGeo-7k, the full pipeline reaches 68%–96% proof accuracy, against 10%–44% for baselines that use neither analogical retrieval nor verifier feedback. Under the same verifier feedback and inference budget, analogical guidance alone lifts GPT-5 from 88% to 96% and o1 from 52% to 80%.

HybridProver joins whole-proof synthesis and stepwise tactic generation in one pipeline: a whole-proof model drafts candidate proofs, and candidates that fail Isabelle's checker are abstracted into proof sketches that keep the high-level structure, then filled subgoal by subgoal by a tactic model and Sledgehammer. On the 244 test problems of the Isabelle version of miniF2F, the pipeline reaches a 73.8% success rate, above the previous best Isabelle result of 61.9%. Both 7B-scale models are trained with supervised fine-tuning and reinforcement learning rewarded by Isabelle verification, indicating that lightweight models can generate Isabelle/HOL proofs.

HybridProver joins whole-proof synthesis and stepwise tactic generation in one pipeline: a whole-proof model drafts candidate proofs, and candidates that fail Isabelle's checker are abstracted into proof sketches that keep the high-level structure, then filled subgoal by subgoal by a tactic model and Sledgehammer. On the 244 test problems of the Isabelle version of miniF2F, the pipeline reaches a 73.8% success rate, above the previous best Isabelle result of 61.9%. Both 7B-scale models are trained with supervised fine-tuning and reinforcement learning rewarded by Isabelle verification, indicating that lightweight models can generate Isabelle/HOL proofs.

This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.

This review systematically surveys the progress, challenges, and prospects of AI for Mathematics (AI4Math), organizing the field into problem-specific modeling (guiding intuition, constructing counterexamples, formal reasoning in closed systems) and general-purpose modeling (natural language reasoning, formal reasoning, mathematical information retrieval), and reports the authors' own PKU undergraduate and PhD qualifying exam evaluations: GPT-4 averages below 60 while reasoning-enhanced models such as o1, DeepSeek-R1, o3-mini, and Gemini 2.5 Pro mostly exceed 90, with o3-mini averaging 84.4 on 58 PhD qualifying exam problems, while research-level mathematics remains an open challenge.